Approximate cloning of structured pure states
Abstract
The no-cloning theorem is a cornerstone result in quantum mechanics that forbids copying of general quantum information. More quantitatively, even given N copies of an unknown state, any channel producing an (N+1)-copy state incurs a nonzero trace-distance error on some inputs. Turning this around, one can ask about the sample complexity of approximate N→ N+1 cloning: for a desired small error ε, what N suffices? For arbitrary pure states on a Hilbert space H, the answer is N≃ dim H/ε - astronomically large for most H of interest. What if the input is promised to lie in a structured family? We examine a range of families fundamental to many-body physics and quantum information: n-mode fermionic Gaussian and Slater states, n-mode bosonic Gaussian states, and qudit phase states. For fermions, we construct optimal cloning channels, reducing the complexity from exponential to polynomial in n. For bosonic Gaussian states, we construct an explicit cloner that certifies polynomial complexity, without any constraint on the energy of the state; this channel, however, is not optimal in general. The fermionic and bosonic results follow from a single representation-theoretic framework, which generalizes previous work by Werner and by Chiribella and Yang; it also yields the cloners' unitary implementation. Furthermore, we explain why for many families the cloning complexity is linear in the dimension of the family manifold, via a controlled saddle-point evaluation of the frame potential at high N. The same calculation accounts for the analogous scaling of tomography complexity. The above framework does not cover phase states; for these we give an independent construction of a channel with optimal cloning fidelity. Our work complements recent results on cloning stabilizer states.
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BibTeX
@misc{herasymenko2026approximate,
title = {{Approximate cloning of structured pure states}},
author = {Yaroslav Herasymenko and Junqiao Lin and Philip Verduyn Lunel and Dmitry Grinko},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06723},
eprint = {2610.06723},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05783705v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Approximate cloning of structured pure states
AU - Yaroslav Herasymenko
AU - Junqiao Lin
AU - Philip Verduyn Lunel
AU - Dmitry Grinko
PY - 2026
DA - 2026/10/05
PB - HAL
DO - 10.48550/arxiv.2610.06723
UR - https://hal.science/hal-05783705v1
AB - The no-cloning theorem is a cornerstone result in quantum mechanics that forbids copying of general quantum information. More quantitatively, even given N copies of an unknown state, any channel producing an (N+1)-copy state incurs a nonzero trace-distance error on some inputs. Turning this around, one can ask about the sample complexity of approximate N→ N+1 cloning: for a desired small error ε, what N suffices? For arbitrary pure states on a Hilbert space H, the answer is N≃ dim H/ε - astronomically large for most H of interest. What if the input is promised to lie in a structured family? We examine a range of families fundamental to many-body physics and quantum information: n-mode fermionic Gaussian and Slater states, n-mode bosonic Gaussian states, and qudit phase states. For fermions, we construct optimal cloning channels, reducing the complexity from exponential to polynomial in n. For bosonic Gaussian states, we construct an explicit cloner that certifies polynomial complexity, without any constraint on the energy of the state; this channel, however, is not optimal in general. The fermionic and bosonic results follow from a single representation-theoretic framework, which generalizes previous work by Werner and by Chiribella and Yang; it also yields the cloners' unitary implementation. Furthermore, we explain why for many families the cloning complexity is linear in the dimension of the family manifold, via a controlled saddle-point evaluation of the frame potential at high N. The same calculation accounts for the analogous scaling of tomography complexity. The above framework does not cover phase states; for these we give an independent construction of a channel with optimal cloning fidelity. Our work complements recent results on cloning stabilizer states.
ER -